cp's OEIS Frontend

A180602 a(n) = (2^(n+1) - 1)^n.

%I A180602 #23 Jun 19 2025 20:54:26
%S A180602 1,3,49,3375,923521,992436543,4195872914689,70110209207109375,
%T A180602 4649081944211090042881,1227102111503512992112190463,
%U A180602 1291749870339606615892191271170049,5429914198235566686555216227881787109375
%N A180602 a(n) = (2^(n+1) - 1)^n.
%C A180602 More generally, we have the following identities:
%C A180602 (1) Sum_{n>=0} m^n* F(q^n*x)^b* log( F(q^n*x) )^n/n! = Sum_{n>=0} x^n* [y^n] F(y)^(m*q^n + b);
%C A180602 (2) Sum_{n>=0} m^n* q^(n^2)* exp(b*q^n*x)*x^n/n! = Sum_{n>=0} (m*q^n + b)^n*x^n/n! for all q, m, b.
%C A180602 This sequence results from (2) when q=2, m=2, b=-1.
%C A180602 For n >= 2, a(n) is the first number in a set of three powerful numbers in arithmetic progression with difference a(n)*(2^n - 1). - _Arkadiusz Wesolowski_, Aug 26 2013
%F A180602 E.g.f.: Sum_{n>=0} 2^(n^2+n) * exp(-2^n*x) * x^n/n!.
%e A180602 E.g.f: A(x) = 1 + 3*x + 7^2*x^2/2! + 15^3*x^3/3! + 31^4*x^4/4! +...
%e A180602 A(x) = exp(-x) + 2^2*exp(-2*x)*x + 2^6*exp(-4*x)*x^2/2! + 2^12*exp(-8*x)*x^3/3! +...
%p A180602 A180602:=n->(2^(n+1) - 1)^n: seq(A180602(n), n=0..10); # _Wesley Ivan Hurt_, Oct 09 2014
%t A180602 Table[(2^(n + 1) - 1)^n, {n, 0, 10}] (* _Wesley Ivan Hurt_, Oct 09 2014 *)
%o A180602 (PARI) {a(n)=n!*polcoeff(sum(k=0, n, 2^(k^2+k)*exp(-2^k*x+x*O(x^n))*x^k/k!), n)}
%o A180602 (Magma) [(2^(n+1)-1)^n : n in [0..11]]; // _Arkadiusz Wesolowski_, Aug 26 2013
%o A180602 (Python)
%o A180602 def A180602(n): return ((1<<n+1)-1)**n # _Chai Wah Wu_, Sep 13 2024
%Y A180602 Cf. A086459 (signed, offset 1), variants: A055601, A079491, A136516, A165327.
%Y A180602 Cf. A001694.
%K A180602 nonn,easy
%O A180602 0,2
%A A180602 _Paul D. Hanna_, Sep 11 2010
%E A180602 Name changed by _Arkadiusz Wesolowski_, Aug 26 2013